Generalized heterogeneous algebras and partial interpretations

Abstract

A notion of heterogeneous algebras generalizing the concepts of total, partial and regular algebras is introduced consisting of a family of carrier sets, a family of total functions and a family of definedness predicates. Partial interpretations are families of partial functions mapping such generalized algebras homomorphically onto partial heterogeneous algebras. Classes of generalized algebras can be specified equationally by generalized abstract types. This notion of abstract type is particularly well suited for the description of programming languages since the notion of homomorphism between generalized algebras allows to obtain fully abstract models in a uniform way — as weakly terminal models. Sufficient conditions for the existence of initial and terminal models and interpretations are given, the model classes of generalized abstract types is analysed using lattice-theoretic methods and the relationship to the classical concept of partial functions is explained. The main advantage of this approach — a uniform treatment of "strict" and "nonstrict" functions — is shown by an extended example where all recursively enumerable (finite and infinite) sequences of natural numbers are specified as a generalized abstract type with nonstrict basic functions.

This research was partially sponsored by the Sonderforschungsbereich 49, Programmiertechnik, Munich